I cannot remove the solder from my pots even with a 30 W soldering iron... so, instead of reoving the wires this way (for the ultimate purpose of installing new pickups), can i just clip the old wire off close to the existing peice of solder, and then solder the new wire over the top / beside it with some fresh solder? Or will this cause problems...

it will work but the joint may be weak. use some heat shrink wrap over the joint and it should be fine.
i've done it before, it works pretty well. make sure you leave yourself plenty of wire and wrap it nice and sturdy. i used electrical tape, but i imagine shrink wrap would work a lot better.
get a hotter iron and do it properly imo
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